[UPLIFTJEE]

Let $\mathrm{z}_{1}=2+3 \mathrm{i}$ and $\mathrm{z}_{2}=3+4 \mathrm{i}$. The set $\mathrm{S}=\left\{\mathrm{z} \in \mathrm{C}:\left|\mathrm{z}-\mathrm{z}_{1}\right|^{2}-\left|\mathrm{z}-\mathrm{z}_{2}\right|^{2}=\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}\right\}$ represents a [JEE (Main) 25 Jan 2023 S I]
\begin{itemize}
\item[(A)] hyperbola with the length of the transverse axis 7
\item[(B)] straight line with the sum of its intercepts on the coordinate axes equals -18
\item[(C)] hyperbola with eccentricity 2
\item[(D)] straight line with the sum of its intercepts on the coordinate axes equals 14
\end{itemize}

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[UPLIFTJEE]

Solution

$\left|\mathrm{z}-\mathrm{z}_{1}\right|^{2}-\left|\mathrm{z}-\mathrm{z}_{2}\right|^{2}=\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}$

\[ \begin{aligned} & \Rightarrow\left|(\mathrm{x}-2)^{2}+(\mathrm{y}-3)^{2}\right|-\left|(\mathrm{x}-3)^{2}+(\mathrm{y}-4)^{2}\right|=\left|(2-3)^{2}+(3-4)^{2}\right|=2 \\ & \Rightarrow 2 \mathrm{x}+2 \mathrm{y}=14 \Rightarrow \mathrm{x}+\mathrm{y}=7 \text { (straight line) Sum of intercepts is } 14 \end{aligned} \]

[UPLIFTJEE]

Answer

D

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